We study the generation of gravitational waves (GWs) during a cosmological first-order phase transition (PT) using the recently introduced Higgsless approach to numerically simulate the fluid motion induced by the PT. We present for the first time GW spectra sourced by bulk fluid motion in the aftermath of strong first-order PTs (α = 0.5), alongside weak (α = 0.0046) and intermediate (α = 0.05) PTs, previously considered in the literature. We find that, for intermediate and strong PTs, the kinetic energy in our simulations decays, following a power law in time. The decay is potentially determined by non-linear dynamics and hence related to the production of vorticity. We show that the assumption that the source is stationary in time, characteristic of compressional motion in the linear regime (sound waves), agrees with our numerical results for weak PTs, since in this case the kinetic energy does not decay with time. We then provide a theoretical framework that extends the stationary assumption to one that accounts for the time evolution of the source: as a result, the GW energy density is no longer linearly increasing with the source duration, but proportional to the integral over time of the squared kinetic energy fraction. This effectively reduces the linear growth rate of the GW energy density and allows to account for the period of transition from the linear to the non-linear regimes of the fluid perturbations. We validate the novel theoretical model with the results of simulations and provide templates for the GW spectrum for a broad range of PT parameters.